16 research outputs found
On globally sparse Ramsey graphs
We say that a graph has the Ramsey property w.r.t.\ some graph and
some integer , or is -Ramsey for short, if any -coloring
of the edges of contains a monochromatic copy of . R{\"o}dl and
Ruci{\'n}ski asked how globally sparse -Ramsey graphs can possibly
be, where the density of is measured by the subgraph with
the highest average degree. So far, this so-called Ramsey density is known only
for cliques and some trivial graphs . In this work we determine the Ramsey
density up to some small error terms for several cases when is a complete
bipartite graph, a cycle or a path, and colors are available
3‐Color bipartite Ramsey number of cycles and paths
The k-colour bipartite Ramsey number of a bipartite graph H is the least integer n for which
every k-edge-coloured complete bipartite graph Kn,n contains a monochromatic copy of H. The
study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and,
independently, by Gy´arf´as and Lehel, who determined the 2-colour Ramsey number of paths. In
this paper we determine asymptotically the 3-colour bipartite Ramsey number of paths and (even)
cycles
Graphs with second largest eigenvalue less than
We characterize the simple connected graphs with the second largest
eigenvalue less than 1/2, which consists of 13 classes of specific graphs.
These 13 classes hint that , where is
the minimum real number for which every real number greater than is a
limit point in the set of the second largest eigenvalues of the simple
connected graphs. We leave it as a problem.Comment: 36 pages, 2 table
Monochromatic cycles in 2-edge-colored bipartite graphs with large minimum degree
For graphs , and , write if each
red-blue-edge-coloring of yields a red or a blue . The Ramsey
number is the minimum number such that the complete graph
. In [Discrete Math. 312(2012)], Schelp formulated
the following question: for which graphs there is a constant such
that for any graph of order at least with ,
. In this paper, we prove that for any , if is a
balanced bipartite graph of order with
, then , where
is a matching with edges contained in a connected component. By
Szem\'{e}redi's Regularity Lemma, using a similar idea as introduced by [J.
Combin. Theory Ser. B 75(1999)], we show that for every , there is an
integer such that for any the following holds: Let
such that . Let be a
balanced bipartite graph on vertices with
. Then for each red-blue-edge-coloring
of , either there exist red even cycles of each length in , or there exist blue even cycles of each length
in . Furthermore, the bound
is asymptotically tight. Previous
studies on Schelp's question on cycles are on diagonal case, we obtain an
asymptotic result of Schelp's question for all non-diagonal cases
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete